Nonlinear Principal Components and Long-run Implications of Multivariate Diffusions By

We investigate a method for extracting nonlinear principal components (NPCs). These NPCs maximize variation subject to smoothness and orthog-onality constraints; but we allow for a general class of constraints and multi-variate probability densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these NPCs. By exploiting the theory of continuous-time, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the NPCs maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the NPCs behave as scalar autoregressions with heteroskedastic innovations; this supports semiparametric identification and estimation of a multivariate reversible diffusion process and tests of the overidentifying restrictions implied by such a process from low-frequency data. We also explore implications for stationary, possibly nonreversible diffusion processes. Finally, we suggest a sieve method to estimate the NPCs from discretely-sampled data. 1. Introduction. Principal components are functions of the data that capture maximal variation in some sense. Often they are restricted to be linear functions of the underlying data as in original analyses of Pearson [27] and Hotelling [23]. In this paper we study the extraction of nonlinear principal components (NPCs) using information encoded in the probability density of the data. Formally, the NPCs maximize variation subject to orthogonality and smoothness constraints where smoothness constraints are enforced by a quadratic form f expressed in terms of the gradients of functions. Specifically, the quadratic form is